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118       XIII    INTEGRATION

By virtue of the definitions, it is enough to show that if u e f, v e f and
u g v, then ^(u) <; n*(v). Since - u e ^ it follows that v - u = v + ( M) is
defined throughout X, belongs to / and is 0; hence

0 g p*(v - w) = /i*(t? + ( - M)) = A**(fl) + ju*( - w)

by virtue of (13.5.3).

We shall often write \  fd\L or | */(#) rfju(x) (resp. | fdfior j f(x

instead of /**(/) (resp. ^*(/)).

For any subset A of X, we put ju*(A) = /i*(<pA) and ^*(A) =
where <^)A is the characteristic function of A (12.7). These numbers /**(A)
and ^*(A) are ^0 (possibly H-oo); they are called, respectively, the outer
measure and inner measure of A.

Example

(13.5.11) Let 1 be Lebesgue measure on R and let I = ]#, [ be an open
interval in R. We shall show that /L*(I) == b  a (which is equal to + oo if
&=+oo or<2= oo). If a < a' < V < b, then there exists a continuous
mapping/of R into the unit interval [0, 1], with support contained in [<2, ],

and equal to 1 throughout [a', 6'] (4.5.2). We have (+cf(t)dt^bf-af.

J  oo

Conversely, for each function g e #"R(R) such that 0 ^ g ^ (pl, we have
f + V(0dt^b-a. Hence it follows that A*(I) = b-a.

J  oo

Now let U be any 0/?ew sef in R. Then its connected components (3.19) are
open intervals ((3.19.1) and (3.19.5)), and the set of connected components
is at most denumerable, because each component contains a point of the
denumerable set Q, and the components are pairwise disjoint. Hence if they
are lk = ~\ak, Afc[, then <?>u = Z Wiv anc^ therefore ((12.7.4) and (13.5.4))

k

(13.5.11.1)

6. NEGLIGIBLE FUNCTIONS AND SETS

A mapping/: X - R is said to be negligible (with respect to the measure
ju), or it-negligible, if Ai*(|/|) = 0. Then afis also negligible for all a ^ 0 in R;
and if \g\ < |/|, then g is negligible.exists an>  the